Show transcribed image text 1· Check-digit codes. Check whether or not the following codewords are valid for the code given (a) the UPC number 7-80133-19831-7 (b) the ISBN-10 number 0-87150-334-X 2. The code used to verify passports uses the weighting vector w = (7.3.1, 7,3,1, ) modulo 10, Calculate the check digit A Ξ P-u, for the passport mimber P 1234567 Show al of the steps needed to encrypt and decrypt the ASCTT message "YES" without a modular arithmetic calculator in an RSA cipher with public key(143,7) 4. An RSA cipher uses public key (899, 517). Plaintext messages are translated to 3-digit ASCII and encrypted one symbol at a time. Decipher the received message 157 675 002 736 183 Hint: An online modular arithmetic calculator will be helpful for this one 5. Decide whether or not each set is a group with respect to the indicated operation. If it is a group, just say "yes it is a group", and check whether or not the group is Abelian. If it is not a group, find the first of the four group properties (closed, associative, identity inverses) that does not hold, and show that it does not hold. (a) R-Q under addition (b) the half-open interval (0,1] under multiplication. (c) the set of positive divisors of 10 with respect to GCD (d) the set of surjective functions from Z to Z with respect to composition (e) the set of polynomials in Rr] having constant term 1 nder ultiplication. 6. Part of two finite group operation tables are shown. Can you complete each table? labc d e f 2 7. I.et (G.*) be a multiplicative group. Prove that Vz, y, 2 E G, ry =エz y = z Clearly explain where each group condition is used in your proof. 8. Let (G,) be a multiplicative group. (a) Show that Vr, y E G, (y-ry尸-y-11-ly (b) Show that for all positive integers n, and for all r, y G. (ydzy)" ydzny 9. Show that a multiplicative group (G,.) is Abelian iff Vr, y EG, (ry)y